Statistics of cosmic density profiles from perturbation theory

The joint probability distribution function (PDF) of the density within multiple concentric spherical cells is considered. It is shown how its cumulant generating function can be obtained at tree order in perturbation theory as the Legendre transform of a function directly built in terms of the initial moments. In the context of the upcoming generation of large-scale structure surveys, it is conjectured that this result correctly models such a function for finite values of the variance. Detailed consequences of this assumption are explored. In particular the corresponding one-cell density probability distribution at finite variance is computed for realistic power spectra, taking into account its scale variation. It is found to be in agreement with $\mathrm{\Lambda }$-cold dark matter simulations at the few percent level for a wide range of density values and parameters. Related explicit analytic expansions at the low and high density tails are given. The conditional (at fixed density) and marginal probability of the slope—the density difference between adjacent cells—and its fluctuations is also computed from the two-cell joint PDF; it also compares very well to simulations. It is emphasized that this could prove useful when studying the statistical properties of voids as it can serve as a statistical indicator to test gravity models and/or probe key cosmological parameters.

Figure 1

A graphical representation of the one-dimensional stationary condition $\lambda ={\mathrm{\Psi }}^{\prime }[\rho ]$. There is a maximum value for $\lambda$ that corresponds to a critical value ${\rho }_c$ for $\rho$ defined in Eq. (42).

Figure 2

The PDF of the one-point density. The blue solid line is the numerical integration, the red dashed line the low $\rho$ asymptotic form of Eq. (46); the other lines correspond to the large $\rho$ asymptotic forms proposed in the text: the dark lines correspond to the form (47) and the red lines to the form (48) and the forms are computed at leading order, at next-to-leading, and next-to-next-to-leading order for respectively the dotted, dot-dashed and dashed curves. The plots are given for ${\sigma }^2=0.45$ and a power law index of $n=-1.5$.

Figure 3

Comparison with simulations (top panel) with residuals (bottom panel). The solid line is the theoretical prediction computed for a variance of ${\sigma }_R^2=0.47$ as measured in the simulation, a power law index of $n=-1.576$ and a running parameter $\alpha =0.439$ corresponding to the input linear power spectrum. The measured PDF in the simulation is shown as a band corresponding to its 1–$\sigma$ error bar (but different data points are correlated). The residuals show the ratio of the measured PDF in bins with the predictions (computed in bins as well). The thin red symbols show the comparison when the running parameter is set to zero in the prediction.

Figure 4

Ratio of the one-point density PDF when the running parameter is taken into account over the PDF when it is not. The running model is the same as in the previous plot. The dashed lines are the ratio of the corresponding asymptotic forms in the low and high density regions.

Figure 5

Contour plot of $\phi (\lambda ,\mu )-\lambda$ (left panel) with a finite radius difference $\mathrm{\Delta }R/R=1/10$ and (right panel) with $\mathrm{\Delta }R/R\to 0$. We see that the structure of the critical region, although deformed, is preserved. In both cases, the restriction of $\phi (\lambda ,\mu )$ to $\mu =0$ is precisely the one-cell cumulant generating function considered in Sec. 3.

Figure 6

Contour plot of $\phi (\lambda ,\mu )-\lambda$, from the simulation. It is to be compared with the left panel of Fig. 5.

Figure 7

The slope generating function ${\phi }_s(\mu )$ for finite differences $\mathrm{\Delta }R/R=0.1$ and $\mathrm{\Delta }R/R=0.01$, and in the limit $\mathrm{\Delta }R/R\to 0$. The corresponding curves are respectively in blue, darker blue and black. The vertical dashed lines show the locations of the critical points, ${\mu }_c^{-}$ and ${\mu }_c^{+}$.

Figure 8

The PDF of the slope for $z=1.46$. The bottom panels show the residuals for $z=1.46$, $z=0.97$ and $z=0.65$ from top to bottom.

Figure 9

(Top panel) The conditional profile, $⟨\widehat{s}{⟩}_{\widehat{\rho }(<R_1)}$ as a function of $\widehat{\rho }(<R_1)$. The thick blue solid line is the result of the numerical integration, the thin dashed line the saddle point approximation Eq. (91). We also present the power law approximation case as a thin (red) solid line. It is shown to depart from the exact prediction in the low density region. The agreement between the theory and the measurements near the origin is quite remarkable. The bottom panels show the residuals computed in bins as a function of the density (with a zoomed plot below). Again the thick symbols correspond to the exact calculation, the thin symbols correspond to the power law approximation.

Figure 10

The conditional profile as a function of $R_2$ and for different choices of $\widehat{\rho }(R_1)$ [to which $\widehat{\rho }(R_2)$ is equal to at $R_2=R_1$]. The blue thick solid lines are the results of numerical integrations; the colored thin lines show the $1\text{−}\sigma$ variance about the expectation. The close-by gray lines are the same calculations but using the saddle point approximations of Eqs. (91) and (c5) respectively. Note in particular the smaller variance of the underdense profile near $R_2\sim 0$.

Figure 11

Same quantities as in Fig. 10 measured here in simulations. The solid lines are the theoretical predictions and the points with error bars are the measurements for both the expected value and its variance. The agreement is spectacular in particular for low density constraints.

Figure 12

The likelihood contours at one, three and five sigmas around the reference model $(n_s=-2.5,\nu =3/2)$, drawn from $\sim 11000$ measurements (inner blue contours) and $\sim 2000$ measurements (outer red contours) of the densities in concentric shells of radii 10 and $11\mathrm{Mpc}/h$.

Figure 13

The path line in the $\rho$ complex plane. We superimposed the contour plot of the imaginary part of $\phi (\lambda )-\lambda \widehat{\rho }$ to check that it follows a $\Im [\phi (\lambda )-\lambda \widehat{\rho }]=0$ line. The starting point on the real axis correspond to the saddle point value.

Figure 14

The path line in the $\lambda$ complex plane for the computation of the large density asymptotic forms.

Figure 15

(Left panels) The residuals of the expected density PDF from smoothing scale of $R_1=4$ (top panel) to $R_1=20h^{-1}\mathrm{Mpc}$. This is for $z=0.97$ (same convention as in Fig. 3). The values of ${\sigma }^2$ at the smoothing scale is given in the inset. The value in square brackets is the linear value. (Right panels) The residuals for the expected average profile between scale $R_1$ and $R_2=R_1+1h^{-1}\mathrm{Mpc}$. From top to bottom we have $R_1=6$ to $18h^{-1}\mathrm{Mpc}$. Same convention as in Fig. 8.